Widely-connected sets in the bucket-handle continuum

A connected topological space is said to be widely-connected if each of its non-degenerate connected subsets is dense in the entire space. The object of this paper is the construction of widely-connected subsets of the plane. We give a completely metrizable example that answers a question of Paul Erd\H{o}s and Howard Cook, while a similar example answers a question of Jerzy Mioduszewski.